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Asymptotic Distribution of the Likelihood Function in the Independent not Identically Distributed Case

A. N. Philippou and G. G. Roussas
The Annals of Statistics
Vol. 1, No. 3 (May, 1973), pp. 454-471
Stable URL: http://www.jstor.org/stable/2958104
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Asymptotic Distribution of the Likelihood Function in the Independent not Identically Distributed Case
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Abstract

Let Θ be an open subset of Rk and for each θ ∈ Θ, let X1, ⋯, Xn be independent rv's defined on the probability space (X, A, Pθ), and let pj, θ be the distribution of the rv Xj. Let $f_j(\bullet; \theta)$ be a specified version of the Radon-Nikodym derivative of pj, θ with respect to a σ-finite measure μ and set fj(θ) = fj(Xj; θ). Furthermore, for θ, θ* ∈ Θ, set φj(θ, θ*) = [ fj(θ*)/fj(θ)]1/2 and suppose that φj(θ, θ*) is differentiable in quadratic mean (qm) with respect to θ* at (θ, θ), when the probability measure Pθ is employed, with qm derivative φ̇j(θ). Set Δn(θ) = 2n-1/2 ∑n j=1 φ̇j(θ), Γj(θ) = 4Eθ[φ̇j(θ)φ̇j'(θ)], Γ̄n(θ) = n-1 ∑n j=1 Γj(θ), and suppose that Γ̄n(θ) → Γ̄(θ) and Γ̄(θ) is positive definite on Θ. Finally, for hn → h ∈ Rk, set θn = θ + hnn -1/2 and Λn(θ) = log[ dPn,θn /dPn,θ], where Pn,θ stands for the restriction of Pθ to An = σ(X1, ⋯, Xn). Then, under suitable--and not too hard to verify--conditions, we obtain, the following results. The limits are taken as n → ∞. THEOREM 1. Λn(θ) - h'Δn(θ) → -1/2 h'Γ̄(θ)h in Pθ-probability, θ ∈ Θ. THEOREM 2. $\mathscr{L}\lbrack\Delta_n(\theta) \mid P_\theta\rbrack \Rightarrow N(0, \bar{\Gamma}(\theta)), \theta \in \Theta$. THEOREM 3. $\mathscr{L}\lbrack\Lambda_n(\theta) \mid P_\theta\rbrack \Rightarrow N(-\frac{1}{2} h'\bar{\Gamma}(\theta)h, h'\bar{\Gamma}(\theta)h), \theta \in \Theta$. THEOREM 4. Λn(θ) - h'Δn(θ) → -1/2h'Γ̄(θ)h in Pθn -probability, θ ∈ Θ. THEOREM 5. $\mathscr{L}\lbrack\Lambda_n(\theta) \mid P_{\theta_n}\rbrack \Rightarrow N(\frac{1}{2}h'\bar{\Gamma}(\theta)h, h'\bar{\Gamma}(theta)h), \theta \in \Theta$. THEOREM 6. $\mathscr{L}\lbrack\Delta_n(\theta) \mid P_{\theta_n}\rbrack \Rightarrow N(\bar{\Gamma}(\theta)h, \bar{\Gamma}(\theta)), \theta \in \Theta$.

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